Biassociahedra Revisited Joint work with Samson Saneblidze Ron - - PowerPoint PPT Presentation
Biassociahedra Revisited Joint work with Samson Saneblidze Ron Umble, Millersville U Lehigh Geometry/Topology Conference Honoring Don Davis on the occassion of his 70th birthday May 23, 2015 A-infinity Bialgebras Biassociahedron KK n , m
Joint work with Samson Saneblidze Ron Umble, Millersville U Lehigh Geometry/Topology Conference Honoring Don Davis on the occassion of his 70th birthday May 23, 2015
Biassociahedron KKn,m is a contractible (m + n − 3)-polytope
Biassociahedron KKn,m is a contractible (m + n − 3)-polytope With a single top dimensional cell em+n−3
Biassociahedron KKn,m is a contractible (m + n − 3)-polytope With a single top dimensional cell em+n−3 KKn,m ∼
= KKm,n
Biassociahedron KKn,m is a contractible (m + n − 3)-polytope With a single top dimensional cell em+n−3 KKn,m ∼
= KKm,n
Stasheff’s associahedron Kn = KK1,n
Let C∗ (KK) denote the cellular chains of KKn,m
Let C∗ (KK) denote the cellular chains of KKn,m An A∞-bialgebra is a graded module H with operations
m : H⊗m → H⊗n m,n≥1
and a chain map C∗ (KK) → End (TH) em+n−3 → θn
m
Over a field, H∗ (ΩX) has an induced A∞-bialgebra structure
Over a field, H∗ (ΩX) has an induced A∞-bialgebra structure Extends the A∞-algebra structure discovered by Kadeishvili
Over a field, H∗ (ΩX) has an induced A∞-bialgebra structure Extends the A∞-algebra structure discovered by Kadeishvili Transfer DG bialgebra structure along a cycle-selecting map
H∗ (ΩX) → S∗ (ΩX)
Cells of KKn,m are indexed by m-in/n-out directed graphs
Cells of KKn,m are indexed by m-in/n-out directed graphs em+n−3 ←
→ n outputs m inputs
Cells of KKn,m are indexed by m-in/n-out directed graphs em+n−3 ←
→ n outputs m inputs
{Vertices} ↔ {Certain binary graphs}
Cells of KKn,m are indexed by m-in/n-out directed graphs em+n−3 ←
→ n outputs m inputs
{Vertices} ↔ {Certain binary graphs} When n = 1, graphs are uprooted trees
Compositions in End (TH) are “fraction products”
Compositions in End (TH) are “fraction products” If g f = 0
Compositions in End (TH) are “fraction products” If g f = 0
# outputs from each factor of f = # of factors in g
Compositions in End (TH) are “fraction products” If g f = 0
# outputs from each factor of f = # of factors in g # inputs to each factor of g = # of factors in f
Compositions in End (TH) are “fraction products” If g f = 0
# outputs from each factor of f = # of factors in g # inputs to each factor of g = # of factors in f
{Binary graphs} ↔
{Compositions involving mult and comult }
KK2,2
KK2,3
Our construction of KKn,m has four steps:
Our construction of KKn,m has four steps:
Insert subdivision vertices into the permutahedron Pm+n−2
Our construction of KKn,m has four steps:
Insert subdivision vertices into the permutahedron Pm+n−2 Extend the vertex poset structure to subdivision vertices
Our construction of KKn,m has four steps:
Insert subdivision vertices into the permutahedron Pm+n−2 Extend the vertex poset structure to subdivision vertices Use poset structure to propagate subdivision to higher cells
Our construction of KKn,m has four steps:
Insert subdivision vertices into the permutahedron Pm+n−2 Extend the vertex poset structure to subdivision vertices Use poset structure to propagate subdivision to higher cells Project to an appropriate quotient
Our construction of KKn,m has four steps:
Insert subdivision vertices into the permutahedron Pm+n−2 Extend the vertex poset structure to subdivision vertices Use poset structure to propagate subdivision to higher cells Project to an appropriate quotient
2011 construction inserts too many subdivision vertices, e.g.
z = is not an admissible vertex of KK4,4
Our construction of KKn,m has four steps:
Insert subdivision vertices into the permutahedron Pm+n−2 Extend the vertex poset structure to subdivision vertices Use poset structure to propagate subdivision to higher cells Project to an appropriate quotient
2011 construction inserts too many subdivision vertices, e.g.
z = is not an admissible vertex of KK4,4
The goal of this talk is to identify the issue and indicate a fix
P1 = ∗
P1 = ∗ P2 = I
P1 = ∗ P2 = I P3 is a hexagon
P4 is a solid truncated octahedron
P4 is a solid truncated octahedron Pn is an (n − 1)-dim’l contractible polytope
Let n = {1, 2, . . . n}
{Cells of Pn in codim p} ↔ {Partitions U1| · · · |Up+1 of n}
↔ {Planar trees with p + 1 levels and n + 1 leaves}
Let n = {1, 2, . . . n}
{Cells of Pn in codim p} ↔ {Partitions U1| · · · |Up+1 of n}
↔ {Planar trees with p + 1 levels and n + 1 leaves}
{Vertices of Pn} ↔ Sn ↔ {Binary trees with n levels}
Let n = {1, 2, . . . n}
{Cells of Pn in codim p} ↔ {Partitions U1| · · · |Up+1 of n}
↔ {Planar trees with p + 1 levels and n + 1 leaves}
{Vertices of Pn} ↔ Sn ↔ {Binary trees with n levels} P1 : ∗ ↔ 1 ↔
Let n = {1, 2, . . . n}
{Cells of Pn in codim p} ↔ {Partitions U1| · · · |Up+1 of n}
↔ {Planar trees with p + 1 levels and n + 1 leaves}
{Vertices of Pn} ↔ Sn ↔ {Binary trees with n levels} P1 : ∗ ↔ 1 ↔ P2 : edge ↔ 12 ↔
{vertices} ↔ {1|2, 2|1} ↔
P3
1|2|3 1|3|2 3|1|2 2|1|3 2|3|1 3|2|1 1|23 3|12 13|2 23|1 2|13 12|3
Pm × Pn embeds in Pm+n via em × en → em| (en + m)
Pm × Pn embeds in Pm+n via em × en → em| (en + m) Embedding of P1 × P2 in P3 :
1 × 1|2 → 1|2|3 1 × 2|1 → 1|3|2 1 × 12 → 1|23
Pm × Pn embeds in Pm+n via em × en → em| (en + m) Embedding of P1 × P2 in P3 :
1 × 1|2 → 1|2|3 1 × 2|1 → 1|3|2 1 × 12 → 1|23
(m, n) -shuffles on vertices of Pm × Pn generate vertices of Pm+n
Pm × Pn embeds in Pm+n via em × en → em| (en + m) Embedding of P1 × P2 in P3 :
1 × 1|2 → 1|2|3 1 × 2|1 → 1|3|2 1 × 12 → 1|23
(m, n) -shuffles on vertices of Pm × Pn generate vertices of Pm+n {Vertices of P3} = shuff (1, 2|3) ∪ shuff (1, 3|2) =
{1|2|3, 2|1|3, 2|3|1} ∪ {1|3|2, 3|1|2, 3|2|1}
C∗ (Pn) denotes the cellular chains of Pn
C∗ (Pn) denotes the cellular chains of Pn ∆P : C∗ (Pn) → C∗ (Pn) ⊗ C∗ (Pn)
C∗ (Pn) denotes the cellular chains of Pn ∆P : C∗ (Pn) → C∗ (Pn) ⊗ C∗ (Pn) S-U agrees with A-W on P1 and P2 :
C∗ (Pn) denotes the cellular chains of Pn ∆P : C∗ (Pn) → C∗ (Pn) ⊗ C∗ (Pn) S-U agrees with A-W on P1 and P2 :
∆P () = ⊗
C∗ (Pn) denotes the cellular chains of Pn ∆P : C∗ (Pn) → C∗ (Pn) ⊗ C∗ (Pn) S-U agrees with A-W on P1 and P2 :
∆P () = ⊗ ∆P
⊗ + ⊗
Define ∆(0) P
:= Id and ∆(n)
P
:=
∆(n−1)
P
Define ∆(0) P
:= Id and ∆(n)
P
:=
∆(n−1)
P Denote the up-rooted n-leaf corolla by n
Define ∆(0) P
:= Id and ∆(n)
P
:=
∆(n−1)
P Denote the up-rooted n-leaf corolla by n Think of ∆(n−1) P
(m+1) as a subcomplex of (Pm)×n
Define ∆(0) P
:= Id and ∆(n)
P
:=
∆(n−1)
P Denote the up-rooted n-leaf corolla by n Think of ∆(n−1) P
(m+1) as a subcomplex of (Pm)×n
X n m =
P
(m+1)
Define ∆(0) P
:= Id and ∆(n)
P
:=
∆(n−1)
P Denote the up-rooted n-leaf corolla by n Think of ∆(n−1) P
(m+1) as a subcomplex of (Pm)×n
X n m =
P
(m+1)
x ∈ X n m ↔ n × 1 matrix of up-rooted trees with m + 1 leaves
∆(1) P () = ⊗ implies X 2 1 =
∆(1) P () = ⊗ implies X 2 1 =
P
⊗ + ⊗ implies X 2
2 =
,
∆(1) P () = ⊗ implies X 2 1 =
P
⊗ + ⊗ implies X 2
2 =
,
∈ X 4
3 ⊂ (P3)x4
Y m n =
P
n+1 is a subposet of (Sn)×m
Y m n =
P
n+1 is a subposet of (Sn)×m
y ∈Y m n ↔1 × m matrix of down-rooted trees with n + 1 leaves
Y m n =
P
n+1 is a subposet of (Sn)×m
y ∈Y m n ↔1 × m matrix of down-rooted trees with n + 1 leaves ∆(2) P () = ⊗ ⊗ implies Y 3 1 = {[ ]}
Y m n =
P
n+1 is a subposet of (Sn)×m
y ∈Y m n ↔1 × m matrix of down-rooted trees with n + 1 leaves ∆(2) P () = ⊗ ⊗ implies Y 3 1 = {[ ]} ∆(1) P
⊗ + ⊗ implies Y 2
2 =
,
Y m n =
P
n+1 is a subposet of (Sn)×m
y ∈Y m n ↔1 × m matrix of down-rooted trees with n + 1 leaves ∆(2) P () = ⊗ ⊗ implies Y 3 1 = {[ ]} ∆(1) P
⊗ + ⊗ implies Y 2
2 =
,
∈ Y 4
3
Trees with levels factor uniquely as matrix products of levels
Trees with levels factor uniquely as matrix products of levels x ∈ X n m factors uniquely as a matrix product x = x1 · · · xm
∈ X 2
2
Trees with levels factor uniquely as matrix products of levels x ∈ X n m factors uniquely as a matrix product x = x1 · · · xm
∈ X 2
2 y ∈ Y m n
factors uniquely as a matrix product y = yn · · · y1 ∈ Y 2
2
A bisequence matrix of graphs αyi xj has the form
αy1
x1
· · · αy1
xp
. . . . . . αyq
x1
· · · αyq
xp
A bisequence matrix of graphs αyi xj has the form
αy1
x1
· · · αy1
xp
. . . . . . αyq
x1
· · · αyq
xp
A transverse product of bisequence matrices has form
αy1
p
. . . αyq
p
x1
· · · βq
xp
p · · · αyq p
βq
x1 · · · βq xp
A typical Block Transverse Product (BTP) has the form
α1
2
α5
2
α4
2
α3
2
α1
1
α5
1
α4
1
α3
1
β3
1
β3
2
β3
3
β1
1
β1
2
β1
3
= α1
2
α5
2
α4
2
β3
1
β3
2
α1
1
α5
1
α4
1
β3
3
2
β1
1
β1
2
1
β1
3
A typical Block Transverse Product (BTP) has the form
α1
2
α5
2
α4
2
α3
2
α1
1
α5
1
α4
1
α3
1
β3
1
β3
2
β3
3
β1
1
β1
2
β1
3
= α1
2
α5
2
α4
2
β3
1
β3
2
α1
1
α5
1
α4
1
β3
3
2
β1
1
β1
2
1
β1
3
The BTP acts associatively on bisequence matrices
Vertices of KKn,m “generated by” X n m−1 × Y m n−1 ⊂ Pm+n−2
Vertices of KKn,m “generated by” X n m−1 × Y m n−1 ⊂ Pm+n−2 X 2 1 × Y 2 1 =
Vertices of KKn,m “generated by” X n m−1 × Y m n−1 ⊂ Pm+n−2 X 2 1 × Y 2 1 =
X 2 2 × Y 3 1 =
A = [aij] is {, }-matrix; rows contain exactly once
A = [aij] is {, }-matrix; rows contain exactly once B = [bij] is {, }-matrix; columns contain exactly once
A = [aij] is {, }-matrix; rows contain exactly once B = [bij] is {, }-matrix; columns contain exactly once (A, B) is an (i, j)-edge pair if
aij ai+1,j bij bi,j+1 =
A = [aij] is {, }-matrix; rows contain exactly once B = [bij] is {, }-matrix; columns contain exactly once (A, B) is an (i, j)-edge pair if
aij ai+1,j bij bi,j+1 =
,
A = [aij] is {, }-matrix; rows contain exactly once B = [bij] is {, }-matrix; columns contain exactly once (A, B) is an (i, j)-edge pair if
aij ai+1,j bij bi,j+1 =
,
Not an edge pair:
,
If (A, B) is an (i, j)-edge pair...
If (A, B) is an (i, j)-edge pair... Ai∗ is the matrix obtained from A by deleting the ith row
If (A, B) is an (i, j)-edge pair... Ai∗ is the matrix obtained from A by deleting the ith row B∗j is the matrix obtained from B by deleting the jth column
If (A, B) is an (i, j)-edge pair... Ai∗ is the matrix obtained from A by deleting the ith row B∗j is the matrix obtained from B by deleting the jth column The (i, j)-transposition of AB is B∗jAi∗
If (A, B) is an (i, j)-edge pair... Ai∗ is the matrix obtained from A by deleting the ith row B∗j is the matrix obtained from B by deleting the jth column The (i, j)-transposition of AB is B∗jAi∗ Ex:
(1,1)-transpose
⇒ [] []
If (A, B) is an (i, j)-edge pair... Ai∗ is the matrix obtained from A by deleting the ith row B∗j is the matrix obtained from B by deleting the jth column The (i, j)-transposition of AB is B∗jAi∗ Ex:
(1,1)-transpose
⇒ [] []
If c = C1 · · · Cr and (Ck, Ck+1) is an (i, j)-edge pair, define
T k
ij (c) := C1 · · · C ∗j k+1C i∗ k · · · Cr
Iterate T on X n m−1 × Y m n−1 in all possible ways:
Zn,m =
itjt · · · T k1 i1j1 (u)
m−1 × Y m n−1
Iterate T on X n m−1 × Y m n−1 in all possible ways:
Zn,m =
itjt · · · T k1 i1j1 (u)
m−1 × Y m n−1
X n
m−1 × Y m n−1 Zn,m
Iterate T on X n m−1 × Y m n−1 in all possible ways:
Zn,m =
itjt · · · T k1 i1j1 (u)
m−1 × Y m n−1
X n
m−1 × Y m n−1 Zn,m Vertices of KK2,3 :
Iterate T on X n m−1 × Y m n−1 in all possible ways:
Zn,m =
itjt · · · T k1 i1j1 (u)
m−1 × Y m n−1
X n
m−1 × Y m n−1 Zn,m Vertices of KK2,3 :
Iterate T on X n m−1 × Y m n−1 in all possible ways:
Zn,m =
itjt · · · T k1 i1j1 (u)
m−1 × Y m n−1
X n
m−1 × Y m n−1 Zn,m Vertices of KK2,3 :
Our 2011 construction admits all elements of Zn,m
Our 2011 construction admits all elements of Zn,m If m + n ≥ 8, certain elements of Zn,m must be discarded
Our 2011 construction admits all elements of Zn,m If m + n ≥ 8, certain elements of Zn,m must be discarded Consider the following vertex u ∈ X 4 3 × Y 4 3 :
=
T 3 1,1 (u) =
=
z = T 2 1,1T 3 1,1 (u) =
= ∈ Z4,4
z = T 2 1,1T 3 1,1 (u) =
= ∈ Z4,4
The 2nd matrix above is not a bisequence matrix, however...
∃! balanced factorization with bisequence indecomposable
factors ABC =
[ ]
∃! balanced factorization with bisequence indecomposable
factors ABC =
[ ]
A vertex is admissible if balanced factorization is ∆P-coherent
∃! balanced factorization with bisequence indecomposable
factors ABC =
[ ]
A vertex is admissible if balanced factorization is ∆P-coherent We now realize that the product AB fails to be ∆P-coherent
Write the entries of AB in their balanced factorizations:
AB =
=
Write the entries of AB in their balanced factorizations:
AB =
=
Up-rooted trees with same row (input) leaf sequences are
Write the entries of AB in their balanced factorizations:
AB =
=
Up-rooted trees with same row (input) leaf sequences are
is not an edge of ∆P (12|3) ⊂ (P2 × P1)×2
Write the entries of AB in their balanced factorizations:
AB =
=
Up-rooted trees with same row (input) leaf sequences are
is not an edge of ∆P (12|3) ⊂ (P2 × P1)×2
Since AB fails to be ∆P-coherent, vertex ABC is discarded